Recalling the orthogonality relation of Bessel functions on a
finite disk of radius \(R\)  [1]},
\(\int _{0}^{R}J_{\nu }\left(\frac{z_{vm}r}{R}\right)J_{\nu }\left(\frac{z_{vn}r}{R}\right)r dr = & \nonumber \\=\frac{R^{2}}{2}\delta _{nm}\left[J_{\left|\nu \right|+1}\left(z_{vm}\right)\right]^{2},&\) 
